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Modeling Heterogeneous Variance–Covariance Components in Two-Level Models

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  • George Leckie
  • Robert French
  • Chris Charlton
  • William Browne

Abstract

Applications of multilevel models to continuous outcomes nearly always assume constant residual variance and constant random effects variances and covariances. However, modeling heterogeneity of variance can prove a useful indicator of model misspecification, and in some educational and behavioral studies, it may even be of direct substantive interest. The purpose of this article is to review, describe, and illustrate a set of recent extensions to two-level models that allow the residual and random effects variance–covariance components to be specified as functions of predictors. These predictors can then be entered with random coefficients to allow the Level-1 heteroscedastic relationships to vary across Level-2 units. We demonstrate by simulation that ignoring Level-2 variability in residual variances leads the Level-1 variance function regression coefficients to be estimated with spurious precision. We discuss software options for fitting these extensions, and we illustrate them by reanalyzing the classic High School and Beyond data and two-level school effects models presented by Raudenbush and Bryk.

Suggested Citation

  • George Leckie & Robert French & Chris Charlton & William Browne, 2014. "Modeling Heterogeneous Variance–Covariance Components in Two-Level Models," Journal of Educational and Behavioral Statistics, , vol. 39(5), pages 307-332, October.
    • Handle: RePEc:sae:jedbes:v:39:y:2014:i:5:p:307-332
    DOI: 10.3102/1076998614546494
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    References listed on IDEAS

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    1. Murray Aitkin, 1987. "Modelling Variance Heterogeneity in Normal Regression Using GLIM," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 36(3), pages 332-339, November.
    2. Browne, William J. & Draper, David & Goldstein, Harvey & Rasbash, Jon, 2002. "Bayesian and likelihood methods for fitting multilevel models with complex level-1 variation," Computational Statistics & Data Analysis, Elsevier, vol. 39(2), pages 203-225, April.
    3. Hedeker, Donald & Nordgren, Rachel, 2013. "MIXREGLS: A Program for Mixed-Effects Location Scale Analysis," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 52(i12).
    4. Harvey, A C, 1976. "Estimating Regression Models with Multiplicative Heteroscedasticity," Econometrica, Econometric Society, vol. 44(3), pages 461-465, May.
    5. Leckie, George, 2014. "runmixregls: A Program to Run the MIXREGLS Mixed-Effects Location Scale Software from within Stata," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 59(c02).
    6. Wei Shen & Thomas A. Louis, 1998. "Triple‐goal estimates in two‐stage hierarchical models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(2), pages 455-471.
    7. Murray Aitkin, 1999. "A General Maximum Likelihood Analysis of Variance Components in Generalized Linear Models," Biometrics, The International Biometric Society, vol. 55(1), pages 117-128, March.
    8. Youngjo Lee & John A. Nelder, 2006. "Double hierarchical generalized linear models (with discussion)," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 55(2), pages 139-185, April.
    9. Charles E. McCulloch & John M. Neuhaus, 2011. "Prediction of Random Effects in Linear and Generalized Linear Models under Model Misspecification," Biometrics, The International Biometric Society, vol. 67(1), pages 270-279, March.
    10. David J. Spiegelhalter & Nicola G. Best & Bradley P. Carlin & Angelika Van Der Linde, 2002. "Bayesian measures of model complexity and fit," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(4), pages 583-639, October.
    11. Harvey Goldstein & Sally Thomas, 1996. "Using Examination Results as Indicators of School and College Performance," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 159(1), pages 149-163, January.
    12. Donald Hedeker & Robin J. Mermelstein & Hakan Demirtas, 2008. "An Application of a Mixed-Effects Location Scale Model for Analysis of Ecological Momentary Assessment (EMA) Data," Biometrics, The International Biometric Society, vol. 64(2), pages 627-634, June.
    13. Browne, William J., 2006. "MCMC algorithms for constrained variance matrices," Computational Statistics & Data Analysis, Elsevier, vol. 50(7), pages 1655-1677, April.
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    3. Pablo-Romero, María del P. & Sánchez-Braza, Antonio, 2017. "The changing of the relationships between carbon footprints and final demand: Panel data evidence for 40 major countries," Energy Economics, Elsevier, vol. 61(C), pages 8-20.
    4. Steele, Fiona & Clarke, Paul & Leckie, George & Allan, Julia & Johnston, Derek, 2017. "Multilevel structural equation models for longitudinal data where predictors are measured more frequently than outcomes: an application to the effects of stress on the cognitive function of nurses," LSE Research Online Documents on Economics 64893, London School of Economics and Political Science, LSE Library.
    5. Fiona Steele & Paul Clarke & George Leckie & Julia Allan & Derek Johnston, 2017. "Multilevel structural equation models for longitudinal data where predictors are measured more frequently than outcomes: an application to the effects of stress on the cognitive function of nurses," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 180(1), pages 263-283, January.

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